Generalized methods and solvers for noise removal from piecewise constant signals. II. New methods

Abstract

Removing noise from signals which are piecewise constant (PWC) is a challenging signal processing problem that arises in many practical scientific and engineering contexts. In the first paper (part I) of this series of two, we presented background theory building on results from the image processing community to show that the majority of these algorithms, and more proposed in the wider literature, are each associated with a special case of a generalized functional, that, when minimized, solves the PWC denoising problem. It shows how the minimizer can be obtained by a range of computational solver algorithms. In this second paper (part II), using this understanding developed in part I, we introduce several novel PWC denoising methods, which, for example, combine the global behaviour of mean shift clustering with the local smoothing of total variation diffusion, and show example solver algorithms for these new methods. Comparisons between these methods are performed on synthetic and real signals, revealing that our new methods have a useful role to play. Finally, overlaps between the generalized methods of these two papers and others such as wavelet shrinkage, hidden Markov models, and piecewise smooth filtering are touched on.

An Information-Based Approach to Change-Point Analysis with Applications to Biophysics and Cell Biology

“…BIC is already known to be an asymptotic result that is applicable only at large N (22), but it is particular poorly suited to the change-point problem because the BIC complexity is too small for small N and much too large for large N, and therefore it is difficult to recommend this approach under any circumstance. Little, Jones, and co-workers (1,5,23,24) have recently introduced a number of convex methods closely related to change-point analysis. Although convexity is clearly a desirable property of an algorithm, the mathematical meaning of the convexified optimization is less clear.…”

“…The approach we discuss in this article is called ''change-point analysis'' and has been applied widely (1)(2)(3)(4)(5), including previous applications to single-molecule biophysics problems (6)(7)(8)(9)(10). We have recently developed an information-based approach to model selection, one that is new to our knowledge: the frequentist information criterion (FIC), an approach that greatly simplifies the analysis (C. H. LaMont and P. A. Wiggins, unpublished).…”

Iterative Potts and Blake–Zisserman minimization for the recovery of functions with discontinuities from indirect measurements

“…In (2.13), we have to solve a J-jump sparsity problem for A = id of the form Nonetheless, the method has quadratic complexity with respect to the data size L. As in the case of the Potts problem with A = id, there are greedy algorithms which may perform well, in practice, but which do not guarantee a global optimum; see [7,8] for a discussion.…”

“…Furthermore, there are strategies to prune the search space which speed up the computations in practice [48,49]. Besides the dynamic programming approach there are greedy algorithms which perform well in practice but which come without theoretical guarantees; for a discussion, see [7,8].…”

An efficient solution to sparse linear prediction analysis of speech

“…The optimization problem is thus iteratively approaching the solution for the ideal l 0 -norm objective function. We also mention that, an interesting review is made in [9,10], on several solvers for the general problem of mixed l p -l 0 -norm minimization in the context of piece-wise constant function approximation, which indeed their adaptation to the problem of sparse linear prediction analysis can be beneficial (particularly the stepwise jump penalization algorithm, which is shown to be highly efficient and reliable in detection of sparse events).…”